= Solution
For $X\subset\mathbb A^n$ with vanishing ideal $I(X)$, the <Zariski tangent space> at $P$ is
$$
\boxed{T_PX=\{v\in k^n:\sum_{j=1}^n\partial_jf(P)v_j=0\ \hbox{for all }f\in I(X)\}.}
$$
It is the kernel of the <Jacobian matrix> of generators for the ideal, or intrinsically the dual of $\mathfrak m_P/\mathfrak m_P^2$. A point is a <singular point of an algebraic variety> when this tangent dimension exceeds the local dimension of the variety; otherwise it is nonsingular. For an irreducible affine variety the local dimension at a closed point equals its dimension.
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